thomas5267
  • thomas5267
Prove that the polynomials \(P_n\) has n distinct root for all n. \(P_n\) are characteristic polynomials of a particular type of matrix. \[ P_n=-xA_{n-1}-\frac{1}{2}A_{n-2}\\ A_n = \left( \dfrac{-x+\sqrt{x^2-1}}{2}\right)^n + \left(\dfrac{-x-\sqrt{x^2-1}}{2}\right)^n \] \(A_n\) satisfies the following recurrence relation: \[ A_n=-x A_{n-1}-\frac{1}{4}A_{n-2}\\ A_1=-x\\ A_2=x^2-\frac{1}{2} \]
Mathematics
katieb
  • katieb
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anonymous
  • anonymous
im not sure if we need to find th characteristic coefficient and show they are distinct
anonymous
  • anonymous
@zzr0ck3r
zzr0ck3r
  • zzr0ck3r
no idea

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thomas5267
  • thomas5267
I want to prove these matrices are diagonalisable. \[ M_5=\begin{pmatrix} 0&\frac{1}{2}&0&0&0\\ 1&0&\frac{1}{2}&0&0\\ 0&\frac{1}{2}&0&\frac{1}{2}&0\\ 0&0&\frac{1}{2}&0&1\\ 0&0&0&\frac{1}{2}&0 \end{pmatrix}\\ M_7=\begin{pmatrix} 0&\frac{1}{2}&0&0&0&0&0\\ 1&0&\frac{1}{2}&0&0&0&0\\ 0&\frac{1}{2}&0&\frac{1}{2}&0&0&0\\ 0&0&\frac{1}{2}&0&\frac{1}{2}&0&0\\ 0&0&0&\frac{1}{2}&0&\frac{1}{2}&0\\ 0&0&0&0&\frac{1}{2}&0&1\\ 0&0&0&0&0&\frac{1}{2}&0 \end{pmatrix}\\ \] In order words, \((M_n)_{i,i+1}=(M_n)_{i+1,i}=\frac{1}{2}\) except for \((M_n)_{2,1}=(M_n)_{n-1,n}=1\)
thomas5267
  • thomas5267
The old question has gotten so convoluted I figured that I will post a new one.
zzr0ck3r
  • zzr0ck3r
This is just an algorithm...
anonymous
  • anonymous
seems diagonalizable , but look at the diagonal zero which might have zero and its determinant might be zero in both and not being singular not sure though i can't help. maybe @Empty
zzr0ck3r
  • zzr0ck3r
find the eigen values and the eigen vectors
thomas5267
  • thomas5267
As far as I can check, from \(M_3\) to \(M_{15}\) the matrices all have distinct eigenvalues.
zzr0ck3r
  • zzr0ck3r
ahh you are trying to show for the whole family?
thomas5267
  • thomas5267
Yes. Favard's theorem seems to apply in here. Those polynomials should be orthogonal.
anonymous
  • anonymous
well, there is a reason why i hate orthogonality. i'll watch and learn!
anonymous
  • anonymous
@dan815 @Kainui @Empty when ever your free just lets try on this!
thomas5267
  • thomas5267
I know nothing about Favard's theorem but I found this seemingly useful theorem on google.
thomas5267
  • thomas5267
I have no idea whether \(A_n\) is orthogonal or not. It does not seem like the case.
Empty
  • Empty
Can you use the Cayley Hamilton theorem? It says that every matrix satisfies its characteristic equation.
thomas5267
  • thomas5267
What I want to prove is that the matrices \(M_n\) is always diagonalisable. A sufficient but not necessary condition is that the characteristic polynomials of \(M_n\) have n distinct roots. Not sure how Cayley Hamilton Theorem will help though.

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